An algorithmic approach to small limit cycles of nonlinear differential systems: The averaging method revisited. (March 2023)

• Record Type: Journal Article
• Authors: Huang, Bo; Yap, Chee
• Is Part Of: Journal of symbolic computation. Volume 115(2023)
• Page Start: 492
• Page End: 517
• Publication Date: 2023-03
• Subjects: Algorithmic approach -- Averaging method -- Center -- Limit cycle -- Nonlinear differential systems; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2020.09.001
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: This paper introduces an algorithmic approach to the analysis of bifurcation of limit cycles from the centers of nonlinear continuous differential systems via the averaging method. We develop three algorithms to implement the averaging method. The first algorithm allows one to transform the considered differential systems to the normal form of averaging. Here, we restricted the unperturbed term of the normal form of averaging to be identically zero. The second algorithm is used to derive the computational formulae of the averaged functions at any order. The third algorithm is based on the first two algorithms and determines the exact expressions of the averaged functions for the considered differential systems. The proposed approach is implemented in Maple and its effectiveness is shown by several examples. Moreover, we report some incorrect results in published papers on the averaging method.